Subsection 2.1.1 Unit-factor Method
The unit-factor method or the unity bracket method is a widely used method for unit conversions using the rules of algebra. This method utilizes the fact that any number or expression can be multiplied by one (unity) without changing its value. The advantage of this method is that it explicitly accounts for units, rather than treating a conversion factor as a magic number.
To use this method, conversion factors are expressed as fraction equal to one, and multiplied against the original value while keeping careful track of the units in each term. Multiplying a value by a unit-factor and canceling any dimensional units appearing in both the numerator and denominator will express the original quantity in different units.
For example, since \(\inch{1} = \cm{2.54}\text{,}\) dividing both sides by \(\inch{1}\) inch gives the unit-factor
\begin{equation*}
1 = \left[\frac{\cm{2.54}}{\inch{1}}\right]\text{.}
\end{equation*}
The fraction bar in this term can be read as “per”, so this unit-factor says that there are “2.54 centimeters per inch”.
To convert \(36\) inches to cm, just multiply by \(36\) inches by the appropriate unit-factor, and cancel the inch units that appear in both the numerator and denominator.
\begin{equation*}
\inch{36} \times \left[\frac{\cm{2.54} }{\inch{1}}\right] = 36 \cancel{\text{ in}} \times \left[\frac{2.54 \text{ cm}}{\cancel{1 \text{ in}}}\right] = \cm{91.44}
\end{equation*}
To convert from cm to inches, use the reciprocal of this unit-factor, which is also unity.
\begin{equation*}
\cm{100} \times \left[\frac{\inch{1}}{\cm{2.54} }\right] = 100 \cancel{\text{ cm}} \times \left[\frac{1 \text{ in}}{2.54 \cancel{\text{ cm}}}\right] = \inch{39.37}
\end{equation*}
Thus \(\yd{1}\) (\(\inch{36}\)) is a bit shorter than a meter (\(\cm{100}\)), and \(\m{1}\) is a little more than \(\inch{39}.\)
Unit conversion is often easier within the the SI system, since the relations between units are logical, and the prefixes increase or decrease by \(10^3\text{,}\) so unit conversions can often be done in your head.
Several different conversion factors can be applied sequentially or simultaneously until only the desired set of dimensional units remains. This way, you only need to remember a few common conversion factors to make many different conversions. Here are some relations that you should remember:
\begin{align*}
\mi{1} \amp = \ft{5280}\\
\ft{1} \amp = \inch{12}\\
\yd{1} \amp = \ft{3}\\
\inch{1} \amp = \cm{2.54}\\
\m{1} \amp= \cm{100}\\
\hr{1} \amp = \minute{60}\\
\minute{1} \amp = \second{60}
\end{align*}
With these values we can find the conversion factor between miles per hour and meters per second, as shown in the next example.
More conversion factors available in
Figure 2.1.3 or online.
Example 2.1.1. Distance Conversion.
How many feet are there in a kilometer?
Answer.
\begin{equation*}
d = \ft{3208}
\end{equation*}
Solution.
Refer to
Table 2.2.3 for the length relations needed for the unit-factors.
\begin{align*}
d \amp= \km{1} \times
\left[\frac{\m{1000}}{\km{1}}\right]
\left[\frac{\ft{1}}{\m{.3048}}\right]\\
\amp = \ft{3208}
\end{align*}
Example 2.1.2. Speed Conversion.
Express 10 miles per hour (mph) in units of meters per second (m/s) using a sequence of unit-factors.
Determine a single conversion factor to change from mph to m/s.
Answer.
\begin{equation*}
10 \mathrm{\ mph} = \mps{4.47}
\end{equation*}
\begin{equation*}
\mps{0.447} = 1 \mathrm{\ mph}
\end{equation*}
Solution.
\begin{align*}
10 \mathrm{\ mph}\amp = \frac{\mi{10}}{\hr{1}}\times
\left[\frac{\ft{5280}}{\mi{1}}\right]
\left[\frac{\inch{12}}{\ft{1}}\right]
\left[\frac{\cm{2.54}}{\inch{1}}\right]
\left[\frac{\m{1}}{\cm{100}}\right] \times
\left[\frac{\hr{1}}{\minute{60}}\right]
\left[\frac{\minute{1}}{\second{60}}\right]\\
\amp = \frac{\mi{10}}{\hr{1}}\times
\left[ \frac{\m{1609.3}}{\mi{1}} \right]\times
\left[ \frac{\hr{1}}{\second{3600}} \right]\\
\amp = \frac{\mi{10}}{\hr{1}}\times
\left[ \frac{\mps{0.447}}{\mph{1}} \right]\\
\amp = \mps{4.47}
\end{align*}
By simplifying the unit-factors, we see that
\begin{equation*}
\mps{0.447} =\mph{1}\text{.}
\end{equation*}
